Unified Modelling Theory for Qubit Representation

using Quantum Field Graph ModelsVishal Sahni

Quantum-Nano Computing Systems CentreDayalbagh Educational Institute (Deemed University)

Dayalbagh, Agra INDIA

Indo-US Workshop on System of Systems Engineering

Indian Institute of Technology Kanpur

October 26-28, 2009

• Linear graph theory, a branch of topology, has been applied to such diverse systems as ranging from electrical networks through real physical systems and “conceptual” socio-economic-environmental systems to creational systems.

• The approach was developed at MSU and pioneered by Prof. Herman E. Koenig, father of Physical System Theory.

• The vast success of Physical System Theory as an operational modelling methodology has been attributed to the fact that it is not just based on analogies but invokes the fundamental properties of instrumentation in identifying an appropriate pair of complementary terminal variables (across and through variables) in each discipline of its application.

• It is founded on linear graph theory in capturing the structure as a model of interrelationships between parts of the whole, which is the essence of systems thinking.

• Linear graph theory represents one step towards a systems modelling discipline which coordinates various branches of knowledge into one scientific order.

(Satsangi 2006).

Table Examples of systems and complementary variables

(Satsangi, 2008)

Single Qubit

Figure 1

Superposition of two independent qubits

Figure 2

( ) ( )1 1 1 2 2 20 1 ; 0 1ψ α β ψ α β≡ + ≡ +

NOT Gate Operation

Figure 3

Hadamard Gate Operation

Figure 4.1

Controlled NOT Gate

Swap Gate

Figure 6

Bell (Entangled States)

Figure 7

Quantum Teleportation

Figure 8

Stages in Quantum Teleportation

After Alice conveys her measurement CC to Bob, there are 4 conditions

Summary

Theory of ‘Many Things’• In terms of systems modelling, instead of the system scientists and

practitioners getting bogged down with similar dreams of modelling theory of everything, one could perhaps look for ‘modelling theory of many things’.

• There is certainly a possibility of founding such a unified modelling theory of many things, not everything, based on linear graph theory and unified field theory, not just quantum or string theory but perhaps some futuristic M-string theory or whatever that qualifies as the grand unified theory of so-called everything.

• This ‘modelling theory of many things’, will then span perhaps all kinds of systems, including natural systems, designed physical systems, designed abstract systems and human-activity systems [8].

High-End Complexity Systems• All kinds of systems can give rise to their own peculiar problems, which can be

attempted to be resolved to the extent possible.

• Kristy Kitto [1] prepared a complexity scale for systems ranging fromSimple (e.g. projectile motion, billiard balls, thermodynamic equilibrium,

microeconomics etc. amenable to Newtonian mechanics, thermodynamics, computational complexity, algorithmic information theory etc.)through

Complicated (e.g. weather dynamics, food webs etc. amenable to chaos / fractals, statistical mechanics, catastrophe theory, network theory etc.)

toComplex (e.g. bound states, quantum tunneling, electron and photon

behaviour, genetic regulatory networks, quark and gluon behaviour, biological development, evolution of mind, language, societies . . . at the high-end amenable to quantum field theory, evolution and natural selection, post modernism etc.)

• It is at this higher end of complexity scale that the unified quantum field graph theory holds considerable promise and potential for modelling systems successfully [8].

Kitto’s Complexity Scale

Quantum field graph theory approach, which appears only as an alternative in the present work, would emerge as the modelling tool of choice therein.

Conclusions

A modelling theory founded on linear graph theory and quantum force-fields is indeed capable of modelling a large variety of Quantum Information Processing systems.

References[1] Kitto Kristy, “High end Complexity”, International Journal of General Systems, Vol. 37, No. 6, Dec.

2008, 689-714.[2] Sahni V., “Quantum Computing from a Linear Graph Theoretic Perspective”, Seventh Canadian

Summer School on Quantum Information, IQC, University of Waterloo, Canada, May 2007.[3] Sahni V., “Quantum Information Systems from a Linear Graph Theoretic Perspective and

Futuristic Applications of Quantum Technology”, Proc. of 32nd National Systems Conference (NSC 2008), I.I.T. Roorkee (India), December 2008.

[4] Sahni V., Srivastava D.P., Satsangi P.S., “A Quantum Field Graph Model for Qubit Representation”, Fourth Workshop on Theory of Quantum Computation, Communication and Cryptography (TQC 2009), University of Waterloo, Ontario, Canada.

[5] Satsangi P.S., “Linear Graph Theory for Modelling a Variety of Systems”, Int. Conf. on Differential Geometry and Topology in the Perspective of Modern Trends (DGTPMT – 2006), Dayalbagh Educational Institute, Dayalbagh, Agra, February 2006.

[6] Satsangi P.S., “Autobiographical Retrospectives : Generalizing Physical System Theory for Applied Systems Research from “Real” Physical System through “Conceptual” Socio-Economic Environmental Systems to “Complete” (Physical-Mental-Spiritual) Creational Systems”, International Journal of General Systems, Vol. 35, No. 2, pp. 127-167.

[7] Satsangi P.S., “Generalizing Physical Systems Theory for Applied Systems Research”, Invited Keynote Lecture at the Inaugural Session, 32nd National Systems Conference (NSC 2008), I.I.T. Roorkee (India), December 17, 2008.

[8] Satsangi P.S., “Linear Graph Theoretic General Systems Paradigm – a learning system modelling methodology”, Editorial Review, Literary Paritantra (Systems) – An International Journal on Literature and Theory, Forthcoming Vol. 1, Nos. 1 & 2 , Spring 2009.

[9] Srivastava D.P.., “Study of Some Quantum Phenomena (with Special Reference to Teleportation)”, M.Phil. Pre-Dissertation, Dayalbagh Educational Institute, December 2008.

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